论文标题
$ \ mathbb {r}^n $中原始晶格的等式分布
Equidistribution of primitive lattices in $\mathbb{R}^n$
论文作者
论文摘要
我们将$ \ Mathbb {z}^{n} $内部的原始晶格算作$ d $，因为相对于此类晶格的某些参数，它们的covolume倾向于无穷大。这些参数包括一个晶格跨度的亚皮，即其对格拉斯曼尼亚的投影；它的同性恋班级；及其等效类模量重新缩放和旋转，通常称为形状。我们通过允许在参数的空间中进行集合来添加施密特的先前工作，这些参数足以结论这些参数的关节等分。除了原始的$ d $ lattices本身外，我们还考虑了它们在$ \ mathbb {z}^{n} $中的正交补充，并证明等均分配是共同用于原始晶格及其正交补充的。最后，我们针对原始晶格数量的渐近公式包括明确的误差项。
We count primitive lattices of rank $d$ inside $\mathbb{Z}^{n}$ as their covolume tends to infinity, with respect to certain parameters of such lattices. These parameters include, for example, the subsapce that a lattice spans, namely its projection to the Grassmannian; its homothety class; and its equivalence class modulo rescaling and rotation, often referred to as a shape. We add to a prior work of Schmidt by allowing sets in the spaces of parameters that are general enough to conclude joint equidistribution of these parameters. In addition to the primitive $d$-lattices themselves, we also consider their orthogonal complements in $\mathbb{Z}^{n}$, and show that the equidistribution occurs jointly for primitive lattices and their orthogonal complements. Finally, our asymptotic formulas for the number of primitive lattices include an explicit error term.
